---
tag: NPEBABQ49M
kind: concept
name: Group of Lie type
msc: [20G40, 20D06]
sources:
  - url: https://en.wikipedia.org/wiki/Group_of_Lie_type
same_as:
  wikidata: Q1406425
  wikipedia: Group of Lie type
---

## Summary

A finite group of Lie type is the group of points of a reductive algebraic group that are fixed by a Frobenius-type map.
An example is $\mathrm{GL}_n(\F_q)$ inside $\mathrm{GL}_n$ over the algebraic closure of $\F_q$.
These groups are finite analogues of Lie groups, and their simple quotients form most of the families in the classification of finite simple groups.
They include the classical groups over finite fields, the Chevalley groups of exceptional type, and the twisted groups of Steinberg, Suzuki and Ree.

## Explanations

### L1 human

Matrix groups such as invertible matrices or orthogonal matrices make sense over any field, including a finite field with $q$ elements.
Over a finite field these groups are finite.
For example, $\mathrm{SL}_2(\F_2)$ is the group of $2 \times 2$ matrices with entries in $\{0, 1\}$, arithmetic modulo $2$ and determinant $1$; it has $6$ elements and is isomorphic to $S_3$.
Groups built this way, and their twisted relatives, are the groups of Lie type.
Their simple quotients, such as $\mathrm{PSL}_n(\F_q)$, are the main source of finite simple groups.